(Luận văn thạc sĩ) sự tồn tại nghiệm của phương trình vi phân cấp ba với điều kiện biên dạng ba điểm và dạng tích phân

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

P❍❸▼ ❚❍➚ ❚❍❯ ❚❘❆◆●

❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆
❈❻P ❇❆ ❱❰■ ✣■➋❯ ❑■➏◆ ❇■➊◆ ❉❸◆● ❇❆ ✣■➎▼ ❱⑨
❉❸◆● ❚➑❈❍ P❍❹◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

P❍❸▼ ❚❍➚ ❚❍❯ ❚❘❆◆●
❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆
❈❻P ❇❆ ❱❰■ ✣■➋❯ ❑■➏◆ ❇■➊◆ ❉❸◆● ❇❆ ✣■➎▼ ❱⑨
❉❸◆● ❚➑❈❍ P❍❹◆
◆❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ữợ ồ
ề

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔
tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥
r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔
❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố

t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾

❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥
P❤↕♠ ❚❤à ❚❤✉ ❚r❛♥❣
❳→❝
ừ


ừ ữớ ữợ ồ
r ✣➻♥❤ ❍ò♥❣

✐


ớ ỡ
rữợ tr ở ừ ❧✉➟♥ ✈➠♥✱ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣
❜✐➳t ì♥ s➙✉ s➢❝ tợ r ũ ữớ t t t ữợ ❞➝♥
tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥
♥➔②✳
❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❦❤♦❛ ❚♦→♥ ❝ị♥❣ t♦➔♥ t❤➸
❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣❍❙P ❚❤→✐ ◆❣✉②➯♥ ✤➣ tr✉②➲♥ t❤ư ❝❤♦ tỉ✐ ♥❤ú♥❣
❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✈➔ tổ ỳ ỵ õ
õ qỵ tr sốt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳

❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t
rt ữủ sỹ õ õ ỵ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝
❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ❝↔♠
ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✱
♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾
❚→❝ ❣✐↔

P❤↕♠ ❚❤à ❚❤✉ ❚r❛♥❣
✐✐


▼ư❝ ❧ư❝
❚r❛♥❣ ❜➻❛ ♣❤ư
▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ởt số tự ỡ s









ởt số ỵ ❜➜t ✤ë♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✷

❚♦→♥ tû ❋r❡❞❤♦❧♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸

❍➔♠ ●r❡❡♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỹ tỗ t ừ ữỡ tr ợ
t



ỹ tỗ t ừ ữỡ tr ợ
❞↕♥❣ ❜❛ ✤✐➸♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ỹ tỗ t ừ ữỡ tr ♣❤➙♥ ❝➜♣ ❜❛ ✈ỵ✐ ✤✐➲✉
❦✐➺♥ ❜✐➯♥ ❞↕♥❣ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✹

✸✺
✸✻
✐✐✐


ởt số ỵ t tt

R

t số tỹ



t ré♥❣

A⊂B

A

A∪B

❤đ♣ ❝õ❛ ❤❛✐ t➟♣ ❤đ♣

A

✈➔

B

A∩B

❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ❤đ♣

A

✈➔

B

B

t➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣

ker(f )

❤↕t ♥❤➙♥ ❝õ❛

Coker(f )

✤è✐ ❤↕t ♥❤➙♥ ❝õ❛

✷

❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤

❧➔ t➟♣ ❝♦♥ ❝õ❛

✐✈

B

f
f

A

✈➔

B


▼ð ✤➛✉
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❝â ♥❤✐➲✉ ù♥❣ ử tr
ỹ t ỵ tt ❬✶❪✱ ❬✾❪✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ ❜➔✐ t♦→♥ ①➨t ✤ë ✈ã♥❣ ừ ởt
ợ ữủ t t ợ s♦♥❣ s♦♥❣ ❝→❝ ✈➟t ❧✐➺✉ ❦❤→❝ ♥❤❛✉
❬✽❪✱ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ❞á♥❣ ❝❤↔② ❝õ❛ ♠ët ♠➔♥❣ ♠ä♥❣ ❝❤➜t ❧ä♥❣ ♥❤ỵt
tr➯♥ ❜➲ ♠➦t r➢♥✱ ❦❤✐ ♠ët ♠➔♥❣ ♥❤÷ ✈➟② ❝❤↔② ①✉è♥❣ ởt t t
ữợ t ự s ữ ❝õ❛ sù❝ ❝➠♥❣ ❜➲ ♠➦t✱ ❧ü❝ ❤➜♣ ❞➝♥
❝ơ♥❣ ♥❤÷ ✤ë ợt ữỡ tr ừ ở ụ ✤÷đ❝

✤÷❛ ✈➲ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❬✶✶❪✳ ❚r♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ✤â✱
❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✤÷đ❝ ❞➝♥ ✤➳♥ ❝â t❤➸ ð ❞↕♥❣ ❜❛ ✤✐➸♠✱ ❞↕♥❣ t➼❝❤
t
ự sỹ tỗ t ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
❝➜♣ ❜❛ ✤➛② ✤õ ✈ỵ✐ ❝→❝ ❧♦↕✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❦❤→❝ ♥❤❛✉ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉
sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝✳ ❑ÿ t❤✉➟t ❦❤→ ♣❤ê ❜✐➳♥ ✤÷đ❝ sû ❞ư♥❣
✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❧➔ ♣❤÷ì♥❣ ♣❤→♣
tr ữợ ữỡ ❧✐➯♥ tö❝ ❞ü❛ tr➯♥ ✈✐➺❝
✤→♥❤ ❣✐→ t✐➯♥ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤å ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ ♠ët t❤❛♠ sè t❤➯♠ ✈➔♦✱
s❛✉ õ sỷ ử ỵ t ở ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✺❪✳
❈❤ó♥❣ tỉ✐ ✤➣ ❝❤å♥ ❧✉➟♥ ✈➠♥ ✏❙ü tỗ t ừ ữỡ tr
ợ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥✑✳ ▼ö❝ ✤➼❝❤
❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè t q ừ r r
sỹ tỗ t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✤➛② ✤õ✿

y (t) = f (t, y(t), y (t), y (t)),

✶

0 < t < 1,


tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✱ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t ❜❛ ✤✐➸♠
t
ỗ ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔
t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♠ët sè ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t
✤ë♥❣✱ t♦→♥ tû ❋r❡❞❤♦❧♠ ✈➔ ❤➔♠ ●r❡❡♥✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤↕t ✤÷đ❝ ✤→♥❤ ❣✐→ t✐➯♥
♥❣❤✐➺♠ ❝õ❛ ♠ët ❤å ❜➔✐ t♦→♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✤➛② ✤õ

tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣
t➼❝❤ ♣❤➙♥✳ ❙❛✉ ✤â sû ử ỵ t ở ự ởt
số t q sỹ tỗ t




❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ t❤✐➳t ❝❤♦ ❝❤÷ì♥❣ s❛✉✱
✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✵❪✱ ❬✶✸❪✳

✶✳✶ ▼ët số ỵ t ở


T : A → A✳

▼é✐ ♥❣❤✐➺♠

❣å✐ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤

x

ừ ữỡ tr

x = Tx

ữủ

T

ởt số ỵ t ở s ỵ t ỡ
ữủ sỷ ử ờ tr ự sỹ tỗ t↕✐ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛
❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳
✶✳ ✣à♥❤ ỵ t ở t tỷ ợ số

k

ỵ t ở rr ❝❤♦ ❝→❝ t♦→♥ tû ❧✐➯♥ tư❝ tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ ❤ú✉ ❤↕♥
ỵ t ở r t tỷ t tử
tr ởt t ỗ ré♥❣ ✈➔ ❝♦♠♣❛❝t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✭✈æ
❤↕♥ ❝❤✐➲✉✮✳ ✣➙② ởt tờ qt õ ừ ỵ t ở rr
ỵ t ở r t tû ❧✐➯♥ tư❝ ✈➔ ❝♦♠♣❛❝t
tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
◆❣♦➔✐ r❛ ♠ët số ỵ t ở q trồ ữủ sỷ ử
tr ự sỹ tỗ t ừ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤✐

✸


t ữ ỵ r r t tỷ t
tr ởt t ỗ rộ ừ ổ
ũ ợ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❧➼ t❤✉②➳t ❜➟❝ ❇r♦✉✇❡r ✈➔ ❧➼ t❤✉②➳t
❝❤➾ sè ✤✐➸♠ ❜➜t ✤ë♥❣ ❝ơ♥❣ ❧➔ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư q✉❛♥ trồ ữủ ự ử
tr ự sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝
❝ơ♥❣ ♥❤÷ sỹ tỗ t ừ ữỡ tr t

ỵ t ở
t ữỡ tr t✉②➳♥

x = T x.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳
♠❡tr✐❝

(X, d)

✭①❡♠ ❬✶✸❪✮ ❚♦→♥ tû

✤÷đ❝ ❣å✐ ❧➔ ❝♦ ✈ỵ✐ ❤➺ sè

k

T :M ⊆X→X

✭✶✳✶✮
tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥

♥➳✉ ✈➔ ❝❤➾ ♥➳✉

d(T x, T y) ≤ kd(x, y)
✈ỵ✐ ♠å✐

x, y M

ỵ

k

ố



0 k < 1.

ỵ t ở

sỷ r
T : M ⊆ X → M ❧➔ ♠ët →♥❤ ①↕ tø ▼ ✈➔♦ ❝❤➼♥❤ ♥â❀
✭✐✐✮ ▼ ❧➔ t➟♣ ✤â♥❣✱ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ (X, d)❀
✭✐✐✐✮ ❚ ❧➔ ♠ët →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➺ sè k ✳
❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✶✮

❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ x✱ tù❝ ❧➔ T õ t

ởt t ở tr
ỵ t ở õ ỵ q trồ tr t
t tr ự sỹ tỗ t t ừ
ữỡ tr t

ỵ t ở rr
ợ ỵ t ở ỵ t ở
rr ổ r t ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣✱ t✉② ♥❤✐➯♥ ❝→❝

✹

tt ừ ỵ rr ữủ ợ ọ ỡ s ợ ỵ t
ở

ỵ

ỵ t ở rr

sỷ M t rộ ỗ t ừ Rn tr ✤â N ≥ 1 ✈➔

f : M → M ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â ❢ ❝â ♠ët ✤✐➸♠ t ở
ởt ừ ỵ rr →♣ ❞ư♥❣ ✤÷đ❝ ❝❤♦ ❝→❝ →♥❤
①↕ ❧✐➯♥ tư❝ tr➯♥ ❦❤ỉ♥❣ ỳ t sỹ tỗ t↕✐
♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❛ ♣❤↔✐ ①➨t tr➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥
❤➔♠✱ ✤➙② ❧➔ ❦❤æ♥❣ ❣✐↕♥ ❇❛♥❛❝❤ ✈æ t ổ t ử
ỵ ❜➜t ✤ë♥❣ ❇r♦✉✇❡r✳ ✣è✐ ✈ỵ✐ ❝→❝ t♦→♥ tû tr➯♥ ❦❤ỉ♥❣ ổ
t ỵ t ở r ởt rở
ừ ỵ t ✤ë♥❣ ❇r♦✉✇❡r ✤➦❝ ❜✐➺t ❤✐➺✉ q✉↔ ✈➔ ✤÷đ❝ sû ❞ư♥❣
♣❤ê ỵ s ữủ tr tr ữợ

ỵ t ở r
r s tr ởt tờ qt õ ừ ỵ ❜➜t
✤ë♥❣ ❇r♦✉✇❡r ❝❤♦ ❝→❝ t♦→♥ tû ❝♦♠♣❛❝t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ổ
õ ỵ t ở ❙❝❤❛✉❞❡r✳
❚♦→♥ tû ❝♦♠♣❛❝t ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳
T : D(T ) ⊆ X → Y

✭①❡♠ ❬✶✸❪✮ ❈❤♦

X

❧➔ ♠ët t♦→♥ tû✳

❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝ ♥➳✉

T

T

✈➔

Y

❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔

✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ❝♦♠♣❛❝t ❤❛②

→♥❤ ①↕ ♠å✐ t➟♣ ❜à ❝❤➦♥ ✈➔♦ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣

✤è✐✳
❈→❝ t♦→♥ tû ❝♦♠♣❛❝t ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❤➔♠ ♣❤✐
t✉②➳♥✳ ❚❤ü❝ t➳ ❝â ♥❤✐➲✉ ❦➳t q✉↔ ❝❤♦ ❝→❝ t♦→♥ tû ❧✐➯♥ tư❝ tr➯♥

Rn

✤÷đ❝

❝❤✉②➸♥ s❛♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❦❤✐ t❤❛② t❤➳ t➼♥❤ ❧✐➯♥ tö❝ ❜➡♥❣ t➼♥❤
❝♦♠♣❛❝t✳

❱➼ ❞ö ✶✳✶✳✺✳

●✐↔ sû r➡♥❣ t❛ ❝â ❤➔♠ ❧✐➯♥ tư❝

K : [a, b] × [a, b] × [−R, R] → K,
✺


tr♦♥❣ ✤â

−∞ < a < b < +∞, 0 < R < ∞

✈➔

K = R, C✳

❑➼ ❤✐➺✉

M = {x ∈ C([a, b] , K) : x ≤ R} ,
tr♦♥❣ ✤â
❧✐➯♥ tư❝

x = maxa≤s≤b |x(s)|

✈➔

C([a, b] , K)

❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕

x : [a, b] → K.

❳➨t ❝→❝ t♦→♥ tû t➼❝❤ ♣❤➙♥

b

(T x)(t) =

K(t, s, x(s))ds,
a

t

(Sx)(t) =

K(t, s, x(s))ds,

∀t ∈ [a, b] .

a
õ

S, T



ỵ

M



C([a, b] , K)

t tỷ t

ỵ t ở r



M ởt t rộ ỗ õ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✈➔ ❣✐↔
sû T : M → M ❧➔ t♦→♥ tû ❝♦♠♣❛❝t✳ ❑❤✐ ✤â T ❝â t ở
ởt ừ ỵ t ở r ữủ t
ữ ữợ

q M ởt t rộ ỗ t ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ X ✱ ✈➔ ❣✐↔ sû T : M → M ❧➔ t♦→♥ tû ❧✐➯♥ tö❝✳ ❑❤✐ ✤â T õ
t ở
ỵ r õ ự ử q trồ tr ự sỹ
tỗ t ừ ữỡ tr t ợ t số sỹ tỗ t
❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤

ỵ t ở r
ỵ r ởt t ừ ỵ r r ụ
tữớ ữủ sỷ ử ự sỹ tỗ t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ ✤➛② ✤õ✳

✻


ỵ X ổ f : X → X ❧✐➯♥ tö❝
✈➔ ❝♦♠♣❛❝t✳ ◆➳✉ t➟♣

F = {x ∈ X : x = λf (x), ∀λ ∈ [0, 1]}
❜à ❝❤➦♥ t❤➻ f ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳

✶✳✷ ❚♦→♥ tû ❋r❡❞❤♦❧♠
X

❈❤♦

✈➔

Y

❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑➼ ❤✐➺✉

❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳
Im(T )

❚❛

tợ

T L(X, Y )

ữủ ồ

Ker(T ) Coker(T ) = Y \ Im(T ) ❝â sè ❝❤✐➲✉ ❤ú✉

✤â♥❣ tr♦♥❣

F(X, Y )

❧➔ ❦❤æ♥❣ ❣✐❛♥

Y✳

✭①❡♠ ❬✶✸❪✮ ❚♦→♥ tû ❜à ❝❤➦♥

t♦→♥ tû ❋r❡❞❤♦❧♠ ♥➳✉
❤↕♥ ✈➔

X

L(X, Y )

Y✳

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ t♦→♥ tû ❋r❡❞❤♦❧♠ tø

❈❤➾ sè ❝õ❛ t♦→♥ tû ❋r❡❞❤♦❧♠

T✱

❦➼ ❤✐➺✉

Index(T )

X

tợ

Y

ữủ

Index(T ) = dim(Ker(T )) dim(Coker(T )).

▼ët sè t➼♥❤ ❝❤➜t
X, Y, Z

❈❤♦
✐✮ ◆➳✉
✈➔

❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

T1 : X → Y

T2 T1

✈➔

T2 : Y → Z

❜à ❝❤➦♥✱ ✈➔ ❤❛✐ tr♦♥❣ ❜❛ t♦→♥ tû

❧➔ t♦→♥ tû ❋r❡❞❤♦❧♠✱ t❤➻ t♦→♥ tû ❝á♥ ❧↕✐ ❧➔ t♦→♥ tû ❋r❡❞❤♦❧♠✱ ✈➔

Index(T2 ◦ T1 ) = Index(T1 ) + Index(T2 ).
✐✐✮

T1 , T2

F(X, Y )

❧➔ t➟♣ ♠ð tr♦♥❣

L(X, Y )

✈➔

Index : F(X, Y ) → R
❧➔ ❤➔♠ ❤➡♥❣✳

✼


✶✳✸ ❍➔♠ ●r❡❡♥
❍➔♠ ●r❡❡♥ ❝â ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➔✐ t♦→♥ ❣✐→ trà
❜✐➯♥ ✈➔ ❧➔ ❝æ♥❣ ử q trồ r sỹ tỗ t ❞✉② ♥❤➜t ♥❣❤✐➺♠
❝õ❛ ❝→❝ ❜➔✐ t♦→♥✳
❳➨t ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t

L [y(x)] ≡ p0 (x)
n−1

Mi (y(a), y(b)) ≡

dn−1 y
dn y
+
p
(x)
+ ... + pn (x)y = 0,
1
dxn
dxn−1

k
i d y(a)
αk
k

dx

k=0
tr♦♥❣ ✤â

pi (x), i = 0, ...n

✤✐➸♠ t❤✉ë❝

+

k
i d y(b)
βk
k

dx

= 0,

❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥

i = 1, ...n,

✭✶✳✹✮

(a, b), p0 (x) = 0

✈ỵ✐ ♠å✐

(a, b)✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳

✭①❡♠ ❬✶✵❪✮ ❍➔♠

G(x, t)

✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ●r❡❡♥ ❝õ❛

❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ✭✶✳✸✮ ✲ ✭✶✳✹✮ ♥➳✉ ①❡♠ ♥❤÷ ❤➔♠ ❝õ❛ ❜✐➳♥
♠➣♥ ữợ ợ ồ
r

G(x, t)

(a, t)



tự ❧➔✿

♣❤↔✐ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ tr♦♥❣ ✭✶✳✹✮✱ tù❝ ❧➔

x = t, G(x, t)

i = 1, ..., n.

✈➔ t➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤❡♦ ❜✐➳♥

x

tỵ✐ ❝➜♣

❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝

∂ k G(x, t)
∂ k G(x, t)
lim
− lim−

= 0,
x→t+
x→t
∂xk
∂xk
✭✐✈✮ ✣↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣

x = t✱

(t, b)✱

L [G(x, t)] = 0, x ∈ (t, b).

Mi (G(a, t), G(b, t)) = 0,

(n − 2)

♥â t❤ä❛

[a, t) ✈➔ (t, b]✱ G(x, t) ❧➔ ❤➔♠ ❧✐➯♥ tư❝✱ ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tư❝ tỵ✐ ❝➜♣

L [G(x, t)] = 0, x ∈ (a, t);

✭✐✐✐✮ ❚↕✐

x✱

t ∈ (a, b)✿

♥ ✈➔ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ tr➯♥

✭✐✐✮

✭✶✳✸✮

(n − 1)

t❤❡♦ ❜✐➳♥

x

k = 0, ..., n − 2.

❝õ❛

G(x, t)

❧➔ ❣✐→♥ ✤♦↕♥ ❦❤✐

❝ö t❤➸

∂ n−1 G(x, t)
∂ n−1 G(x, t)
1
lim+
−
lim
=
−
.

x→t
x→t−
∂xn−1
∂xn−1
p0 (t)
✽


ỵ s r sỹ tỗ t t ừ r

ỵ

ỗ t t

t t tr







t tr

õ t tữớ t tỗ t

t r tữỡ ự ợ t
t ữỡ tr t t➼♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t

dn y

dn−1 y
L [y(x)] ≡ p0 (x) n + p1 (x) n−1 + ... + pn (x)y = −f (x),
dx
dx

✭✶✳✺✮

✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t❤✉➛♥ ♥❤➜t

n−1

Mi (y(a), y(b)) ≡

k
i d y(a)
αk
k

dx

k=0
tr♦♥❣ ✤â ❝→❝ ❤➺ sè

+

k
i d y(b)
βk
k

dx

= 0,

i = 1, ...n,

✭✶✳✻✮

pj (x) ✈➔ ❝→❝ ❤➔♠ ✈➳ ♣❤↔✐ f (x) tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮

❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝✱ ✈ỵ✐

p0 (x) = 0 tr➯♥ (a, b) ✈➔ Mi

❜✐➸✉ ❞✐➵♥ ❝→❝ ❞↕♥❣ ✤ë❝

❧➟♣ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ ❤➺ sè ❤➡♥❣✳
✣à♥❤ ỵ s t ố q ỳ t ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮
✲ ✭✶✳✻✮ ✈ỵ✐ ❜➔✐ t♦→♥ t❤✉➛♥ ♥❤➜t tữỡ ự

ỵ
ợ

t tr t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣

❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ t❤➻ ❜➔✐ t







õ

t ữợ
b

y(x) =

G(x, t)f (t)dt,
a

tr ✤â G(x, t) ❧➔ ❤➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ t❤✉➛♥ t tữỡ ự
ởt số ử ữợ r ❝→❝❤ ①→❝ ✤à♥❤ ❤➔♠ ●r❡❡♥ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥
❣✐→ trà ❜✐➯♥ ❝ö t❤➸✳

❱➼ ❞ö ✶✳✸✳✹✳

❳➨t ❜➔✐ t♦→♥





u (x) = −ϕ(x),


u(0) = u(1) = 0.
✾

0 < x < 1,
✭✶✳✼✮


r ữủ t ữợ s

G(x, t) =




A1 + A2 x,

0 ≤ x ≤ t ≤ 1,
✭✶✳✽✮



B1 + B2 (1 − x),
tr♦♥❣ ✤â

A1 , A2

✈➔

B1 , B2

❧➔ ❝→❝ ❤➔♠ ❝õ❛

0 ≤ t ≤ x ≤ 1.
t✳

❍➔♠ ●r❡❡♥ ♥➔② t❤ä❛ ♠➣♥

✤✐➲✉ ❦✐➺♥ ✭✐✮✳
❉♦ ❤➔♠ ●r❡❡♥

G(x, t) t❤ä❛ ♠➣♥ ❜➔✐ t♦→♥ ❜✐➯♥ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➯♥ ❜✐➯♥ t❤✉➛♥
A1 = B1 = 0✳ ❉♦ ✤â✱ ❤➔♠ ●r❡❡♥



A2 x, 0 ≤ x ≤ t ≤ 1,
G(x, t) =


B2 (1 − x), 0 ≤ t ≤ x ≤ 1.

♥❤➜t ✭✐✐✮ t❛ s✉② r❛ ✤÷đ❝

❝õ❛ ❜➔✐ t♦→♥ ❧➔

✭✶✳✾✮

✣✐➲✉ ❦✐➺♥ ❧✐➯♥ tư❝ ✭✐✐✐✮ ❝❤♦ t❛ ♣❤÷ì♥❣ tr➻♥❤

B2 (1 − t) − A2 t = 0.

✭✶✳✶✵✮

B2 + A2 = 1.

✭✶✳✶✶✮

❚ø ✤✐➲✉ ❦✐➺♥ ✭✐✈✮ t❛ ✤÷đ❝

❚❛ ❝â t❤➸ t➻♠ ❝→❝ ❤➺ sè

A2 , B2

❜➡♥❣ ❝→❝❤ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✵✮ ✈➔

♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✳ ❑➳t q✉↔ t❛ ✤÷đ❝

A2 = 1 − t, B2 = t.

❚❤❛② ❝→❝ ❤➺ sè t➻♠ ✤÷đ❝ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ t❛ ✤÷đ❝ ❤➔♠ ●r❡❡♥

G(x, t) =





x(1 − t),

0 ≤ x ≤ t ≤ 1,



t(1 − x),

0 ≤ t ≤ x ≤ 1.

✭✶✳✶✷✮

❉♦ ✤â✱ ♥❣❤✐➺♠ ❝õ❛ t ữủ ữợ

1

u(x) =

ử ✶✳✸✳✺✳

G(x, t)ϕ(t)dt.
0

❳➨t ❜➔✐ t♦→♥




u(4) = ϕ(x),

0 < x < 1,
✭✶✳✶✸✮



u(0) = u (0) = u (1) = u (1) = 0.
✶✵


õ r tữỡ ự ợ t ❝â ❞↕♥❣


3
2


− t + t x , 0 ≤ t ≤ x ≤ 1,
6
2
G(t, s) =
3
2


− x + x t , 0 ≤ x ≤ t ≤ 1.
6
2
❉♦ ✤â ♥❣❤✐➺♠ ừ t ữủ ữợ

1

u(x) =

G(x, t)ϕ(t)dt.
0

✶✶

✭✶✳✶✹✮


ữỡ
ỹ tỗ t ừ ữỡ tr
❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥
❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤
ỹ tỗ t ừ ữỡ tr ❝➜♣
❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠
◆ë✐ ❞✉♥❣ tr♦♥❣ ♠ư❝ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✸❪✳ ❳➨t ữỡ
tr ợ ❜❛ ✤✐➸♠✿

y (t) = f (t, y(t), y (t), y (t)),
y(0) = y(a) = y(1) = 0,
❑➼ ❤✐➺✉
tư❝ tr♦♥❣
❱ỵ✐

I

I

[0, 1]✱ C(I)

❧➔ ✤♦↕♥

k = 1, 2, ...✱

0

❦➼ ❤✐➺✉

❤➔♠ ❧✐➯♥ tư❝ tỵ✐ ❝➜♣

k

y
C03 (I)

y

✈ỵ✐ ❝❤✉➞♥

✭✷✳✷✮

❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥

C k (I)
I✱

❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝→❝ ❤➔♠ ❝â ✤↕♦

✈ỵ✐ ❝❤✉➞♥

= max( y 0 , y

0 , ...,

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝→❝ ❤➔♠

y(a) = y(1) = 0; L1 (I)

0 < a < 1.

✭✷✳✶✮

= max {|y(t)| , t ∈ I}✳

tr♦♥❣

k

0 < t < 1,

y (k) 0 ).

y ∈ C 3 (I)

t❤ä❛ ♠➣♥

y(0) =

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ▲❡❜❡s❣✉❡ tr♦♥❣ ■

✶✷


ợ tổ tữớ
t ồ t ợ t sè

λ

y (t) = λf (t, y(t), y (t), y (t)),

0 < t < 1,

✭✷✳✸✮

y(0) = y(a) = y(1) = 0,
✈ỵ✐

✭✷✳✹✮

0 ≤ λ ≤ 1.

❇ê ✤➲ ✷✳✶✳✶✳ ❱ỵ✐ λ = 0✱ t



õ t t

tữớ tỗ t↕✐ ❤➔♠ ●r❡❡♥ G(t, s) t÷ì♥❣ ù♥❣✳
❈❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ tr➯♥ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø t➼♥❤ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ ❝➜♣ ❜❛ t❤✉➛♥ ♥❤➜t✳
❙❛✉ ✤➙② t❛ s➩ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ❤➔♠

r ≤ 3)

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

y

= 0

G(t, s)✳

●✐↔ sû

ur (t), (1 ≤

✈➔ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥

❜✐➯♥✿

u1 (0) = 1,
u2 (0) = 0,
u3 (0) = 0,

u1 (a) = 0,

u1 (1) = 0,

u2 (a) = 1,
u3 (a) = 0,

u2 = 0,
u3 (1) = 1.

❑❤✐ ✤â

t2 a + 1
t + 1,
u1 (t) = −
a
a

t2 − t
u2 (t) = 2
,
a −a

t2 − at
u3 (t) =
.
1−a

∂ 3v
(t − s)2

✱ t❛ ❝â
= 0.
❳➨t ❤➔♠ v(t, s) =
2
∂t3
✣➦t v1 (s) = v(0, s), v2 (s) = v(a, s)✱ v3 (s) = v(1, s)✱

❤❛②

s2
(a − s)2
(1 − s)2
v1 (s) = , v2 (s) =
, v3 (s) =
.
2
2
2
✣➦t

ϕ(t, s) = u1 (t)v1 (s) + u2 (t)v2 (s) + u3 (t)v3 (s)✳

♥❣❤✐➺♠ ❝õ❛

v(a, s)

✈➔

y

= 0

✈ỵ✐

s

❝è ✤à♥❤✳ ❍ì♥ ♥ú❛

ϕ(1, s) = v(1, s)✳
✶✸

❑❤✐ ✤â

ϕ(., s)

❧➔

ϕ(0, s) = v(0, s), ϕ(a, s) =


❚ø t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t✱
s✉② r❛

ϕ(t, s) = v(t, s)✱

tù❝ ❧➔

u1 (t)v1 (s) + u2 (t)v2 (s) + u3 (t)v3 (s) =
❚❛ ①→❝ ✤à♥❤ ❤➔♠
❱ỵ✐

G(t, s)

(t − s)2
,
2

∀(t, s) ∈ I 2 .

♥❤÷ s❛✉✿

0 ≤ s ≤ a✿

G(t, s) =




−u2 (t)v2 (s) − u3 (t)v3 (s), 0 ≤ t ≤ s,



u1 (t)v1 (s),
s≤t≤a


(t2 − t)
(t2 − at)

2

2
1 − a2 − a (a − s) − 1 − a (1 − s) , 0 ≤ t ≤ s,
=
2
2

 t − (a + 1)t + a s2 ,
s ≤ t ≤ a,
a
✈ỵ✐

a ≤ s ≤ 1✿

G(t, s) =




−u3 (t)v3 (s),

a ≤ t ≤ s,



u1 (t)v1 (s) + u2 (t)v2 (s), s ≤ t ≤ 1


(t2 − at)



−
(1 − s)2 ,
a ≤ t ≤ s,
1
1
−
a
=
2
2
2

 t − (a + 1)t + a s2 + (t − t) (a − s)2 , s ≤ t ≤ 1.
a
a2 − a
◆➳✉ ❤➔♠

f : I × R3 → R

❧✐➯♥ tö❝✱ t❤➻

t♦→♥ ✭✷✳✸✮✱ ✭✷✳✹✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

y ∈ C 2 (I)

y ∈ C 3 (I)

❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐

✈➔ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

t➼❝❤ ♣❤➙♥

1

y(t) = λ

G(t, s)f (s, y(s), y (s), y (s))ds.
0

✣à♥❤ ♥❣❤➽❛ t♦→♥ tû t✉②➳♥ t➼♥❤

L : C03 (I) → C(I)
①→❝ ✤à♥❤ ❜ð✐

(Ly)(t) = y (t),
✶✹

∀t ∈ I.

✭✷✳✺✮


❑❤✐ ✤â
❧✐➯♥ tư❝

L

❧➔ t♦→♥ tû ❋r❡❞❤♦❧♠ ✈ỵ✐ ❝❤➾ sè ✵ ✈➔ ❝â ❤➔♠ ♥❣÷đ❝ ❤♦➔♥ t♦➔♥

L−1 ✱

✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝

1
−1

(L h)(t) =

∀t ∈ I.

G(t, s)h(s)ds,
0

❚❛ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû ◆❡♠✐ts❦✐

F

❝õ❛

f

♥❤÷ s❛✉✿

F : C 2 (I) → C(I),
∀t ∈ I.

(F y)(t) = f (t, y(t), y (t), y (t)),

❑❤✐ ✤â ữỡ tr t tữỡ ữỡ ợ ữỡ tr

y = λ(L−1 ◦ F ◦ j)(y),
tr♦♥❣ ✤â

j : C03 (I) → C 2 (I)

❧➔ ♣❤➨♣ ♥❤ó♥❣

✭✷✳✻✮

jy(t) = y(t).

❇ê ✤➲ ✷✳✶✳✷✳ y C01(I) tỗ t m1 > 0 s❛♦ ❝❤♦ |y (t)| ≤ m1
✈ỵ✐ ♠å✐ t ∈ I t❤➻ |y(t)| ≤
❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

m1
2

y(t) =

✈ỵ✐ ♠å✐ t ∈ I ✳
t
0 y

(s)ds

t

2 |y(t)| ≤
2 |y(t)| ≤ m1 ,

1
t y
1

y(t) = −

1

|y (s)| ds +
0

❉♦ ✤â

✈➔

|y (s)| ds =

(s)ds✳

❙✉② r❛

|y (s)| ds.

t

0

∀t ∈ I ✳

❚❤❡♦ ❜ê ✤➲ ✷✳✶✳✶✱ ❝❤ó♥❣ t❛ s➩ t trữớ ủ

0 < 1

ỵ ●✐↔ sû ❤➔♠ f : I × R3 → R tử 0 < 1
tỗ t m > 0✱ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ λ s❛♦ ❝❤♦ y

y ừ



ự
ỗ ừ

t

t t



U = y C03 (I); y

(3)

(3)

m ợ ồ

tỗ t ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠✳


✳ ❑❤✐ ✤â

U

❧➔ t➟♣ ❝♦♥

C03 (I) ✈➔ →♥❤ ①↕ H : [0, 1] × U → C03 (I) ①→❝ ✤à♥❤ ❜ð✐ H(λ, y) =

λL−1 ◦ F ◦ j(y) ởt ỗ t t ✤ë♥❣ ❝õ❛ ♥â ❧➔
♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✮✱ ✭✷✳✹✮✳ ❱✐➺❝ ❧ü❛ ❝❤å♥ ❯ ♥❤÷ tr➯♥ ✤↔♠ ❜↔♦
❝â ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ ❜✐➯♥
♥❤➟♥ ✤÷đ❝✳ ❚❛ ❝â
✈➟②

H(1, .)

∂U

H(0, .) ≡ 0✱

❝õ❛ ❯❀ ❞♦ ✤â

t♦→♥

U

ổ

H(, .) ởt ỗ

✈➔

❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣

H(λ, .)

H(1, .) = L−1 ◦ F ◦ j ✳

❇ð✐

✈➔ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐


❚✐➳♣ t❤❡♦ t❛ s➩ tr➻♥❤ ❜➔② ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ❤➔♠

f

✤➸ ✤↕t ✤÷đ❝ ✤→♥❤

❣✐→ t✐➯♥ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✱ ✭✷✳✹✮✳

▼➺♥❤ ✤➲ ✷✳✶✳✹✳ ❈❤♦ ❤➔♠ f : I × R3 R tử tọ
tỗ t↕✐ r1 > 0 s❛♦ ❝❤♦ pf (t, y, p, 0) > 0 ✈ỵ✐ ♠å✐ |p| > r1 ✈➔ ♠å✐ y ∈

R✳ ❑❤✐ ✤â ♥❣❤✐➺♠ ② ❝õ❛

✭✷✳✸✮✱ ✭✷✳✹✮

t❤ä❛ ♠➣♥ |y (t)| ≤ r1 ✈➔ |y(t)| ≤

r1
2✱

∀t ∈ I ✳
❈❤ù♥❣ ♠✐♥❤✳

|y (t)| ≤ r1
r1 ✳

y = 0

●✐↔ sû

✈ỵ✐ ♠å✐

❑❤✐ ✤â ❤♦➦❝

❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✮✱ ✭✷✳✹✮✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤

t ∈ I ✳ sỷ ữủ tỗ t t1 I

y (t1 ) > r1

❤♦➦❝

y (t1 ) < −r1 ✳

✭tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❧➔♠ t÷ì♥❣ tü✮✱ s✉② r❛

y

t2 ∈ I

❧✐➯♥ tư❝ ♥➯♥ tỗ t

t2 (0, 1)

t

tọ

y (t2 ) > r1 , y (t2 ) = 0

✈➔

s❛♦ ❝❤♦

❳➨t tr÷í♥❣ ❤đ♣

|y (t1 )| >

y (t1 ) > r1

max {|y (t)| ; t ∈ I} > r1 ✳

y (t2 ) = max {|y (t)| ; t ∈ I}✳
y (t2 )y (t2 ) ≤ 0✳

❉♦

◆➳✉

❚❤❡♦ ✤✐➲✉ ❦✐➺♥

✭❍✶✮ t❛ ❝â✿

y (t2 )f (t2 , y(t2 ), y (t2 ), y (t2 )) = y (t2 )f (t2 , y(t2 ), y (t2 ), 0) > 0.
❚❤❡♦ ✭✷✳✸✮ ✈➔

0<λ≤1

s✉② r❛✿

0 ≥ λy (t2 )y (t2 ) > 0.
✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t✳
◆➳✉

t2 = 0✱

tù❝ ❧➔

y

✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t t↕✐

t = 0✱

❦❤✐ ✤â

y (0) ≤ 0

✈➔

y (0) > r1 ✳
◆➳✉

y (0) = 0✱

tø ✭❍✶✮ t❛ ❝â

y (0)y (0) = y (0)f (0, 0, y (0), 0) > 0,
s✉② r❛

y (0) > 0✳

❉♦ ✤â

y

✤ì♥ ✤✐➺✉ t ợ



y (t) > y (0) = 0

ữỡ tỹ ữ tr➯♥

t > 0✳

❦❤ỉ♥❣ t❤➸ ❧➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛

❱➻ ✈➟②

y (0)

y

t

0

✈➔

t > 0✱

✤ì♥ ✤✐➺✉ t➠♥❣ ✈ỵ✐

|y (t)|✳

t

s✉② r❛

❣➛♥

0

✈➔

▼➙✉ t❤✉➝♥

✈ỵ✐ ❣✐↔ t❤✐➳t✳
◆➳✉
t❤✐➳t

y (0) < 0

t❤➻

y

❧ã♠ t↕✐

0✳

❙✉② r❛

y (0) > r1 > 0✳
✶✻

y (0) ≤ 0✳

▼➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔

◆➳✉

t2 = 1 ✱

①➨t t÷ì♥❣ tü t❛ ❝ơ♥❣ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥ ♥❤÷ tr➯♥✳

❉♦ ✤â

y (t) − r1 ≤ 0,

∀t ∈ I.

❚÷ì♥❣ tü✱

y (t) ≤ −r1 ,

∀t ∈ I.

|y (t)| ≤ r1 ,

∀t ∈ I.

❙✉② r❛

◆❤÷ ✈➟② tø ❜ê ✤➲ ✭✷✳✶✳✷✮ t❛ ❝â

|y(t)| ≤ r1 /2,

∀t ∈ I.

▼➺♥❤ ✤➲ ✷✳✶✳✺✳ ❈❤♦ ❤➔♠ f : I × R3 → R tử tọ
tỗ t q ∈ L1(I), Φ +∞
: [0, +∞) → (0, +∞) ❧✐➯♥ tö❝✱ 1/Φ ❦❤↔ t➼❝❤
dσ
> q
Φ(σ)

tr➯♥ ❝→❝ ❦❤♦↔♥❣ ❜à ❝❤➦♥✱
✤✐➲✉ ❦✐➺♥

0

L1 ✱

✈ỵ✐ ♠å✐ ω ∈ R ✈➔ r1 tr♦♥❣

✭❍✶✮✱

r1 r1
∀(t, y, p) ∈ I × − ,
× [−r1 , r1 ] .
2 2

|f (t, y, p, ) q(t)(||),

õ tỗ t↕✐ r2 > 0 ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ λ s❛♦ ❝❤♦ ♥❣❤✐➺♠ ② ❝õ❛ ❜➔✐
r1
t♦→♥ ✭✷✳✸✮✱✭✷✳✹✮ ✈ỵ✐ |y(t)| ≤ , |y (t)| ≤ r1 ✱ ∀t ∈ I t❤ä❛ ♠➣♥ |y (t)| ≤ r2
2

✈ỵ✐ ♠å✐ t ∈ I ✳
❈❤ù♥❣ ♠✐♥❤✳

r1 /2

✈➔

●✐↔ sû

|y (t)| ≤ r1

y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✱✭✷✳✹✮ t❤ä❛ ♠➣♥ |y(t)| ≤

✈ỵ✐

❚ø ✤✐➲✉ ❦✐➺♥ ❝õ❛

∀t ∈ I ✳

Φ✱

r2

t❛ ❝â t❤➸ ①→❝ ✤à♥❤

r2 > 0

❜ð✐

0

❚❛ s➩ ❝❤ù♥❣ tä r➡♥❣

t∈I
✈➔

t❤ä❛ ♠➣♥

|y (t)| ≤ r2

✈ỵ✐ ♠å✐

t ∈ I✳

dσ
> q
Φ(σ)

●✐↔ sû ữủ tỗ t

|y (t)| > r2 y(0) = y(a) = y(1) tỗ t s1 (0, a)

s2 ∈ (a, 1) s❛♦ ❝❤♦ y (s1 ) = 0 = y (s2 ) r tỗ t t ∈ (s1 , +s2 ) t❤ä❛

♠➣♥

y (t) = 0.

◆❤÷ ✈➟② t❛ ❝â
❱➻

y ∈ C 3 (I)

|y (t)| = 0

✈➔

|y (t)| > r2

tỗ t

[1 , 2 ] I ✱

t❤ä❛ ♠➣♥ ♠ët tr♦♥❣ ♥❤ú♥❣

✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②✿
✭✐✮

L1 ✳

y (σ1 ) = 0, y (σ2 ) = r2

✈➔

0 < y (t) < r2 ,
✶✼

∀t ∈ (σ1 , σ2 ).


✭✐✐✮

y (σ1 ) = r2 , y (σ2 ) = 0

✈➔

∀t ∈ (σ1 , σ2 ).

0 < y (t) < r2 ,

✭✐✐✐✮

y (σ1 ) = 0, y (σ2 ) = −r2

✈➔

−r2 < y (t) < 0,

∀t ∈ (σ1 , σ2 ).

✭✐✈✮

y (σ1 ) = −r2 , y (σ2 ) = 0

✈➔

−r2 < y (t) < 0,

∀t ∈ (σ1 , σ2 ).

❚❛ ①➨t tr÷í♥❣ ❤đ♣ ✤➛✉ t✐➯♥ ✭❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ ❧➔♠ t÷ì♥❣ tü✮✳

❚ø ✭✷✳✸✮ t❛ ❝â

|y (t)| = λ|f (t, y(t), y (t), y (t))| ≤ λq(t)Φ(|y (t)|),
❱ỵ✐

t ∈ [σ1 , σ2 ]

✈➔

0<λ≤1

t❤➻

∀t ∈ I.

y (t) ≤ q(t)Φ(y (t))✱

❤❛②

y (t)
≤ q(t),
Φ(y (t))
❉♦ ✤â

σ2
σ1

∀t ∈ [σ1 , σ2 ] .

σ2

y (t)
dt ≤
Φ(y (t))

1

q(t)dt ≤

q(t)dt = q
0

σ1

L1 .

❑❤✐ ✤â t❛ ❝â ❜➜t tự

r2
0

d
q
()

õ t ợ ồ
ữủ

|y (t)| ≤ r2 ,

❑❤✐ ✤â ❜➔✐ t♦→♥

❈❤ù♥❣ ♠✐♥❤✳

❳➨t ❝→❝ tr÷í♥❣ ❤đ♣ ỏ t t

t I.

ỵ f
✭❍✷✮✳

r2 ✳

L1 .

✭✷✳✶✮

: I × R3 → R ❧✐➯♥ tư❝ tọ







tỗ t t t ởt

y ởt ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✮✱ ✭✷✳✹✮✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭❍✶✮
r1
t❛ ❝â |y(t)| ≤

✈➔ |y (t)| ≤ r1 ✈ỵ✐ ♠å✐ t ∈ I ✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭❍✷✮ t❛ ❝â
2
|y (t)| ≤ r2 ✈ỵ✐ ♠å✐ t ∈ I ✳
r1
✣➦t r3 = {|f (t, y, p, w)|; t ∈ I, |y| ≤
, |p| ≤ r1 , |w| ≤ r2 }.
2
❑❤✐ ✤â y (3) ≤ r.✱ ✈ỵ✐ r = max(r1 , r2 , r3 )✳
✣➦t

●✐↔ sû

U = {y ∈ C03 (I); y

(3)

< 1 + r}

✈➔ →♣ ❞ö♥❣ ✣à♥❤ ❧➼ ✷✳✶✳✸ t❛ ❝â

✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

❱➼ ❞ö ✷✳✶✳✼✳

❳➨t ❜➔✐ t♦→♥ ❜✐➯♥✿

y (t) = tey(t) (y (t) − 1)(1 + y (t)2 ),
✶✽

0 < t < 1,

✭✷✳✼✮


y(0) = y(a) = y(1) = 0.
❚❛ ❝â
♠å✐

f (t, y, p, w) = tey (p − 1)(1 + w2 )✳

p > 1✳

❍ì♥ ♥ú❛✱

q(t) = t

❑❤✐ ✤â

Φ(w) = 1 + w2

✈➔

✭✷✳✽✮

pf (t, y, p, 0) > 0

ợ

tọ

ữ t ỵ t tỗ t↕✐ ➼t ♥❤➜t ♠ët
♥❣❤✐➺♠✳
❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❤❛✐ r sỹ tỗ t
ừ t



ỗ t

k0 > 0 tọ f (t, k0 t, k0 , 0) > 0 ✈➔ f (t, −k0 t, −k0 , 0) < 0

t ∈ I✳

✈ỵ✐ ồ

+

c > 0, l L1 (I)

ỗ t

ợ

0
ồ

dz
= +
(z)
w R t õ

ữủ

s ợ ♠å✐

ψ : [0, +∞) → (0, +∞)

(t, y, p) ∈ I × [−k0 , k0 ] × [−k0 , k0 ]

|f (t, y, p, w)| (l(t) + c|w|)(|w|).

ỵ ✷✳✶✳✽✳ ❈❤♦ ❤➔♠ f : I × R3 → R ❧✐➯♥ tö❝ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
✭❈✶✮

✈➔

❑❤✐ ✤â ❜➔✐ t♦→♥

✭❈✷✮✳

❈❤ù♥❣
ữợ







tỗ t t t ởt

ú t ự t ữợ ữ s



f 1 : I ì R3 → R

①→❝ ✤à♥❤ ❜ð✐





max(f (t, k0 t, k0 , 0), f (t, y, p, w)), p > k0 ,




f1 (t, y, p, w) = f (t, y, p, w),
−k0 ≤ p ≤ k0 ,






min(f (t, −k0 t, −k0 , 0), f (t, y, p, w)), p < −k0 .
❳➨t ❤å ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ t❤❛♠ sè

λ✿

y (t) = λf1 (t, y(t), y (t), y (t)),
y(0) = y(a) = y(1) = 0,
ợ

0 < 1.

ữợ

t



0 < t < 1,

✭✷✳✾✮
✭✷✳✶✵✮